Construction of a Dirichlet form on metric measure spaces of controlled geometry

2023/10/22 by Butaev, Almaz, Luo, Liangbing, Shanmugalingam, Nageswari
#31C25 #46E35 #49Q20 #65N55 #Analysis of PDEs (math.AP) #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.2310.14436

Abstract

Given a compact doubling metric measure space X that supports a 2-Poincaré inequality, we construct a Dirichlet form on N^1,2(X) that is comparable to the upper gradient energy form on N^1,2(X). Our approach is based on the approximation of X by a family of graphs that is doubling and supports a 2-Poincaré inequality. We construct a bilinear form on N^1,2(X) using the Dirichlet form on the graph. We show that the Γ-limit E of this family of bilinear forms (by taking a subsequence) exists and that E is a Dirichlet form on X. Properties of E are established. Moreover, we prove that E has the property of matching boundary values on a domain Ω⊆ X. This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form E) on a domain in X with a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects.

Related

• Notions of Dirichlet problem for functions of least gradient in metric measure spaces
• Discrete Approximations of Metric Measure Spaces of Controlled Geometry
• Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient
• Geometry and Analysis of Dirichlet forms
• Analysis of Dirichlet forms on graphs